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Author: Günter Last
Publisher: Cambridge University Press
ISBN: 1107088011
Category : Mathematics
Languages : en
Pages : 315
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Book Description
A modern introduction to the Poisson process, with general point processes and random measures, and applications to stochastic geometry.
Author: Günter Last
Publisher: Cambridge University Press
ISBN: 1107088011
Category : Mathematics
Languages : en
Pages : 315
Get Book
Book Description
A modern introduction to the Poisson process, with general point processes and random measures, and applications to stochastic geometry.
Author: Günter Last
Publisher: Cambridge University Press
ISBN: 1108514901
Category : Mathematics
Languages : en
Pages : 316
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Book Description
The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.
Author: Harpreet S. Dhillon
Publisher: Springer Nature
ISBN: 303102379X
Category : Computers
Languages : en
Pages : 131
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Book Description
This book provides a comprehensive treatment of the Poisson line Cox process (PLCP) and its applications to vehicular networks. The PLCP is constructed by placing points on each line of a Poisson line process (PLP) as per an independent Poisson point process (PPP). For vehicular applications, one can imagine the layout of the road network as a PLP and the vehicles on the roads as the points of the PLCP. First, a brief historical account of the evolution of the theory of PLP is provided to familiarize readers with the seminal contributions in this area. In order to provide a self-contained treatment of this topic, the construction and key fundamental properties of both PLP and PLCP are discussed in detail. The rest of the book is devoted to the applications of these models to a variety of wireless networks, including vehicular communication networks and localization networks. Specifically, modeling the locations of vehicular nodes and roadside units (RSUs) using PLCP, the signal-to-interference-plus-noise ratio (SINR)-based coverage analysis is presented for both ad hoc and cellular network models. For a similar setting, the load on the cellular macro base stations (MBSs) and RSUs in a vehicular network is also characterized analytically. For the localization networks, PLP is used to model blockages, which is shown to facilitate the characterization of asymptotic blind spot probability in a localization application. Finally, the path distance characteristics for a special case of PLCP are analyzed, which can be leveraged to answer critical questions in the areas of transportation networks and urban planning. The book is concluded with concrete suggestions on future directions of research. Based largely on the original research of the authors, this is the first book that specifically focuses on the self-contained mathematical treatment of the PLCP. The ideal audience of this book is graduate students as well as researchers in academia and industry who are familiar with probability theory, have some exposure to point processes, and are interested in the field of stochastic geometry and vehicular networks. Given the diverse backgrounds of the potential readers, the focus has been on providing an accessible and pedagogical treatment of this topic by consciously avoiding the measure theoretic details without compromising mathematical rigor.
Author: J. F. C. Kingman
Publisher: Clarendon Press
ISBN: 0191591246
Category : Mathematics
Languages : en
Pages : 118
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Book Description
In the theory of random processes there are two that are fundamental, and occur over and over again, often in surprising ways. There is a real sense in which the deepest results are concerned with their interplay. One, the Bachelier Wiener model of Brownian motion, has been the subject of many books. The other, the Poisson process, seems at first sight humbler and less worthy of study in its own right. Nearly every book mentions it, but most hurry past to more general point processes or Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general context. This book attempts to redress the balance. It records Kingman's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful, and a few key results often produce surprising consequences.
Author: Philippe Biane
Publisher: Springer
ISBN: 3540494022
Category : Mathematics
Languages : en
Pages : 217
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Book Description
This book contains two of the three lectures given at the Saint-Flour Summer School of Probability Theory during the period August 18 to September 4, 1993.
Author: Richard Durrett
Publisher: Springer
ISBN: 3319456148
Category : Mathematics
Languages : en
Pages : 282
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Book Description
Building upon the previous editions, this textbook is a first course in stochastic processes taken by undergraduate and graduate students (MS and PhD students from math, statistics, economics, computer science, engineering, and finance departments) who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and option pricing. One can only learn a subject by seeing it in action, so there are a large number of examples and more than 300 carefully chosen exercises to deepen the reader’s understanding. Drawing from teaching experience and student feedback, there are many new examples and problems with solutions that use TI-83 to eliminate the tedious details of solving linear equations by hand, and the collection of exercises is much improved, with many more biological examples. Originally included in previous editions, material too advanced for this first course in stochastic processes has been eliminated while treatment of other topics useful for applications has been expanded. In addition, the ordering of topics has been improved; for example, the difficult subject of martingales is delayed until its usefulness can be applied in the treatment of mathematical finance.
Author: Jean-Claude Falmagne
Publisher: McGraw-Hill Science, Engineering & Mathematics
ISBN:
Category : Mathematics
Languages : en
Pages : 296
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Book Description
Designed for undergraduate mathematics students or graduate students in the sciences. This book can be used in a prerequisite course for Statistics (for math majors) or Mathematical Modeling. The first eighteen chapters could be used in a one-quarter course, and the entire text is suitable for a one-semester course.
Author: Jim Pitman
Publisher: Springer Science & Business Media
ISBN: 354030990X
Category : Mathematics
Languages : en
Pages : 257
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Book Description
The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes. There is particular focus on the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes.
Author: Andreas E. Kyprianou
Publisher: Springer Science & Business Media
ISBN: 3642376320
Category : Mathematics
Languages : en
Pages : 461
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Book Description
Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their application appears in the theory of many areas of classical and modern stochastic processes including storage models, renewal processes, insurance risk models, optimal stopping problems, mathematical finance, continuous-state branching processes and positive self-similar Markov processes. This textbook is based on a series of graduate courses concerning the theory and application of Lévy processes from the perspective of their path fluctuations. Central to the presentation is the decomposition of paths in terms of excursions from the running maximum as well as an understanding of short- and long-term behaviour. The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lévy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical tractability. The second edition additionally addresses recent developments in the potential analysis of subordinators, Wiener-Hopf theory, the theory of scale functions and their application to ruin theory, as well as including an extensive overview of the classical and modern theory of positive self-similar Markov processes. Each chapter has a comprehensive set of exercises.
Author: Kiyosi Itô
Publisher: Springer
ISBN: 981100272X
Category : Mathematics
Languages : en
Pages : 43
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Book Description
An extension problem (often called a boundary problem) of Markov processes has been studied, particularly in the case of one-dimensional diffusion processes, by W. Feller, K. Itô, and H. P. McKean, among others. In this book, Itô discussed a case of a general Markov process with state space S and a specified point a ∈ S called a boundary. The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on S \ {a} which is absorbed on reaching the boundary a). The study in this lecture is restricted to a simpler case of the boundary a being a discontinuous entrance point, leaving a more general case of a continuous entrance point to future works. He established a one-to-one correspondence between a recurrent extension and a pair of a positive measure k(db) on S \ {a} (called the jumping-in measure and a non-negative number m
